Tail Approximations for Sum of Randomly Weighted Dependent Random Variables with Applications

There has been much work on the approximation of independent or dependent random variables. But we are not aware of any work giving exact results for the approximation of the sum of randomly weighted random variables. In this paper, we derive results for the randomly weighted sum of dependent random variables. The derived expressions are for the upper bound of distribution function, approximating tail behaviour, limiting tail behaviour, approximating value at risk, the tail value at risk(TVaR) and expected shortfail (ES) of sum of randomly weighted dependent random variables and upper bound of the joint distribution function of randomly weighted dependent random variables. Two examples are given